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arXiv:1001.5002 · Journal of Physics Condensed Matter

Benchmark quantum Monte Carlo calculations of the ground-state kinetic, interaction and total energy of the three-dimensional electron gas

Idoia G. Gurtubay, R. Gaudoin, Jose M. Pitarke

Published Jan 27, 2010• 12 citations
arXivHTMLPDF

Abstract

We report variational and diffusion quantum Monte Carlo ground-state energies of the three-dimensional electron gas using a model periodic Coulomb interaction and backflow corrections for N = 54, 102, 178, and 226 electrons. We remove finite-size effects by extrapolation and we find lower energies than previously reported. Using the Hellman-Feynman operator sampling method introduced in Gaudoin and Pitarke (2007 Phys. Rev. Lett. 99 126406), we compute accurately, within the fixed-node approximation, the separate kinetic and interaction contributions to the total ground-state energy. The difference between the interaction energies obtained from the original Slater-determinant nodes and the backflow-displaced nodes is found to be considerably larger than the difference between the corresponding kinetic energies.

DOI
10.1088/0953-8984/22/6/065501

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We report variational and diffusion quantum Monte Carlo ground-state energies of the three-dimensional electron gas using a model periodic Coulomb interaction and backflow corrections for N = 54, 102, 178, and 226 electrons. We remove finite-size effects by extrapolation and we find lower energies than previously reported. Using the Hellman-Feynman operator sampling method introduced in Gaudoin and Pitarke (2007 Phys. Rev. Lett. 99 126406), we compute accurately, within the fixed-node approximation, the separate kinetic and interaction contributions to the total ground-state energy. The difference between the interaction energies obtained from the original Slater-determinant nodes and the backflow-displaced nodes is found to be considerably larger than the difference between the corresponding kinetic energies.
The variational and diffusion quantum Monte Carlo methods are used to calculate the correlation energy of the paramagnetic three-dimensional homogeneous electron gas at intermediate to high density. Ground-state energies in finite cells are determined using Slater-Jastrow-backflow trial wave functions, and finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Our correlation energies in the thermodynamic limit are more accurate than previous results. The present diffusion quantum Monte Carlo energies together with our recently reported [Phys. Rev. B 105, 245135 (2022)] results at low density, are used to parametrize the correlation energy of the electron gas using a functional form that satisfies the exact asymptotic behavior at high density.
The correlation energy of the homogeneous three-dimensional interacting electron gas is calculated using the variational and fixed-node diffusion Monte Carlo methods, with trial functions that include backflow and three-body correlations. In the high-density regime (rs< 5){(r}_{s}<~5)(rs​< 5) the effects of backflow dominate over those due to three-body correlations, but the relative importance of the latter increases as the density decreases. Since the backflow correlations vary the nodes of the trial function, this leads to improved energies in the fixed-node diffusion Monte Carlo calculations. The effects are comparable to those found for the two-dimensional electron gas, leading to much improved variational energies and fixed-node diffusion energies similar to the released-node energies of Ceperley and Alder.
We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.
Highly accurate results for the homogeneous electron gas (HEG) have only been achieved to date within a diffusion Monte Carlo (DMC) framework. Here, we introduce a recently developed stochastic technique, full configuration interaction quantum Monte Carlo (FCIQMC), which samples the exact wave function expanded in plane-wave Slater determinants. Despite the introduction of a basis-set incompleteness error, we obtain a finite-basis energy, which is significantly and variationally lower than any previously published work for the 54-electron HEG at rs{r}_{s}rs​ === 0.5 a.u., in a Hilbert space of 10108{10}^{108}10108 Slater determinants. At this value of rs{r}_{s}rs​, as well as of 1.0 a.u., we remove the remaining basis-set incompleteness error by extrapolation, yielding results comparable to state-of-the-art DMC backflow energies. In doing so, we demonstrate that it is possible to yield highly accurate results with the FCIQMC method in periodic systems.
Excitations of the two-dimensional electron gas, including many-body effects, are calculated with a variational Monte Carlo method. Correlated sampling is introduced to calculate small energy differences between different excitations. The usual pair-product (Slater-Jastrow) trial wave function is found to lack certain correlations entirely so that backflow correlation is crucial. From the excitation energies calculated here, we determine Fermi-liquid parameters and related physical quantities such as the effective mass and the Land\'e g factor of the system. Our results for the effective mass are compared with previous analytic calculations.
We have studied the spin-polarized three-dimensional homogeneous electron gas using the diffusion quantum Monte Carlo method, with trial wave functions including backflow and three-body correlations in the Jastrow factor, and we have used twist averaging to reduce finite-size effects. Calculations of the pair-correlation function, including the on-top pair density, as well as the structure factor and the total energy, are reported for systems of 118 electrons in the density range rs=0.5{r}_{\mathrm{s}}=0.5rs​=0.5--20 a.u., and for spin polarizations of 0, 0.34, 0.66, and 1. We consider the spin resolution of the pair-correlation function and structure factor, and the energy of spin polarization. We show that a control variate method can reduce the variance when twist averaging, and we have achieved higher accuracy and lower noise than earlier quantum Monte Carlo studies.